Non-physical Jounce Examples in Nature

Question

What are some good examples of jounce in the non-physics arena?

The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena and B) you never hear of too many examples past the second derivative outside of said physical arena.

Jerk is relatively easy to perceive when one slams on the breaks and then lets them go, and the best way to describe jounce is an amusement park ride since you're always being jerked around.

Outside of physics, it's a little hard to come by, so I'd like to see if there are other ways of perceiving higher order derivatives in other disciplines.

Many thanks in advance!

Answer 1

You write of "examples past the second derivative," so I'm going to assume the third derivative is of interest to you. As I noted in a comment, "In determining stability of nonhyperbolic fixed points of one-dimensional discrete dynamical systems, it may be necessary to evaluate the so-called Schwarzian derivative. The formula for the Schwarzian derivative involves the third derivative of the function in question."

To expand on this a bit:

Given differentiable $f:I\to I$ where $I$ is an interval, and $x$ such that $f(x)=x$, it can be shown that $x$ is an asymptotically stable fixed point of $f$ (roughly: if you start near $x$, and iterate $f$, you converge to $x$) if $|f'(x)|\lt1$, and unstable if $|f'(x)|\gt1$. Things are more complicated if $|f'(x)|=1$ and, in the case $f'(x)=-1$, the determining factor is the value of the Schwarzian derivative evaluated at $x$. You can look up the Schwarzian derivative --- you'll find it involves $f'''(x)$.

EDIT: Maybe you'd prefer Wilson, Matthews, Greasham, Will, and Copeland, Application of fourth derivative absorption spectroscopy to protein quantitation during purification, Anal Biochem. 1989 Oct;182(1):141-5.